Abstract

Digital optical phase conjugation (DOPC) is a new technique employed in wavefront shaping and phase conjugation for focusing light through or within scattering media such as biological tissues. DOPC is particularly attractive as it intrinsically achieves a high fluence reflectivity in comparison to nonlinear optical approaches. However, the slow refresh rate of liquid crystal spatial light modulators and limitations imposed by computer data transfer speeds have thus far made it difficult for DOPC to achieve a playback latency of shorter than 200ms and, therefore, prevented DOPC from being practically applied to thick living samples. In this paper, we report a novel DOPC system that is capable of 5.3 ms playback latency. This speed improvement of almost 2 orders of magnitude is achieved by using a digital micromirror device, field programmable gate array (FPGA) processing, and a single-shot binary phase retrieval technique. With this system, we are able to focus through 2.3 mm living mouse skin with blood flowing through it (decorrelation time 30ms) and demonstrate that the focus can be maintained indefinitely—an important technological milestone that has not been previously reported, to the best of our knowledge.

© 2015 Optical Society of America

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References

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    [Crossref]

2015 (4)

J. Yoon, K. Lee, J. Park, and Y. Park, “Measuring optical transmission matrices by wavefront shaping,” Opt. Express 23, 10158–10167 (2015).
[Crossref]

C. Ma, X. Xu, and L. V. Wang, “Analog time-reversed ultrasonically encoded light focusing inside scattering media with a 33,000 × optical power gain,” Sci. Rep. 5, 8896 (2015).
[Crossref]

Y. Liu, P. Lai, C. Ma, X. Xu, A. A. Grabar, and L. V. Wang, “Optical focusing deep inside dynamic scattering media with near-infrared time-reversed ultrasonically encoded (TRUE) light,” Nat. Commun. 6, 5904 (2015).

M. Jang, H. Ruan, I. M. Vellekoop, B. Judkewitz, E. Chung, and C. Yang, “Relation between speckle decorrelation and optical phase conjugation (OPC)-based turbidity suppression through dynamic scattering media: a study on in vivo mouse skin,” Biomed. Opt. Express 6, 72–85 (2015).
[Crossref]

2014 (5)

G. Volpe, G. Volpe, and S. Gigan, “Brownian motion in a speckle light field: tunable anomalous diffusion and selective optical manipulation,” Sci. Rep. 4, 3936 (2014).
[Crossref]

T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy, and S. Gigan, “Controlling light in scattering media noninvasively using the photoacoustic transmission matrix,” Nat. Photonics 8, 58–64 (2014).
[Crossref]

H. Ruan, M. Jang, B. Judkewitz, and C. Yang, “Iterative time-reversed ultrasonically encoded light focusing in backscattering mode,” Sci. Rep. 4, 7156 (2014).
[Crossref]

E. H. Zhou, H. Ruan, C. Yang, and B. Judkewitz, “Focusing on moving targets through scattering samples,” Optica 1, 227–232 (2014).
[Crossref]

C. Ma, X. Xu, Y. Liu, and L. V. Wang, “Time-reversed adapted-perturbation (TRAP) optical focusing onto dynamic objects inside scattering media,” Nat. Photonics 8, 931–936 (2014).
[Crossref]

2013 (5)

B. Judkewitz, Y. M. Wang, R. Horstmeyer, A. Mathy, and C. Yang, “Speckle-scale focusing in the diffusive regime with time reversal of variance-encoded light (TROVE),” Nat. Photonics 7, 300–305 (2013).
[Crossref]

H. Yilmaz, W. L. Vos, and A. P. Mosk, “Optimal control of light propagation through multiple-scattering media in the presence of noise,” Biomed. Opt. Express 4, 1759–1768 (2013).
[Crossref]

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[Crossref]

B. Jayet, J.-P. Huignard, and F. Ramaz, “Optical phase conjugation in Nd: YVO4 for acousto-optic detection in scattering media,” Opt. Lett. 38, 1256–1258 (2013).
[Crossref]

P. Lai, Y. Suzuki, X. Xu, and L. V. Wang, “Focused fluorescence excitation with time-reversed ultrasonically encoded light and imaging in thick scattering media,” Laser Phys. Lett. 10, 075604 (2013).
[Crossref]

2012 (4)

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012).
[Crossref]

D. B. Conkey, A. M. Caravaca-Aguirre, and R. Piestun, “High-speed scattering medium characterization with application to focusing light through turbid media,” Opt. Express 20, 1733–1740 (2012).
[Crossref]

Y. M. Wang, B. Judkewitz, C. A. DiMarzio, and C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat. Commun. 3, 928 (2012).
[Crossref]

P. Lai, X. Xu, H. Liu, and L. V. Wang, “Time-reversed ultrasonically encoded optical focusing in biological tissue,” J. Biomed. Opt. 17, 0305061 (2012).
[Crossref]

2011 (2)

X. Xu, H. Liu, and L. V. Wang, “Time-reversed ultrasonically encoded optical focusing into scattering media,” Nat. Photonics 5, 154–157 (2011).
[Crossref]

D. Akbulut, T. J. Huisman, E. G. van Putten, W. L. Vos, and A. P. Mosk, “Focusing light through random photonic media by binary amplitude modulation,” Opt. Express 19, 4017–4029 (2011).
[Crossref]

2010 (3)

C.-L. Hsieh, Y. Pu, R. Grange, and D. Psaltis, “Digital phase conjugation of second harmonic radiation emitted by nanoparticles in turbid media,” Opt. Express 18, 12283–12290 (2010).
[Crossref]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[Crossref]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[Crossref]

2008 (1)

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2, 110–115 (2008).
[Crossref]

2007 (1)

2004 (1)

2003 (1)

D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14–25 (2003).
[Crossref]

1997 (1)

Akbulut, D.

Blu, T.

Boccara, A. C.

T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy, and S. Gigan, “Controlling light in scattering media noninvasively using the photoacoustic transmission matrix,” Nat. Photonics 8, 58–64 (2014).
[Crossref]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[Crossref]

Bossy, E.

T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy, and S. Gigan, “Controlling light in scattering media noninvasively using the photoacoustic transmission matrix,” Nat. Photonics 8, 58–64 (2014).
[Crossref]

Bromberg, Y.

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[Crossref]

Caravaca-Aguirre, A. M.

Chaigne, T.

T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy, and S. Gigan, “Controlling light in scattering media noninvasively using the photoacoustic transmission matrix,” Nat. Photonics 8, 58–64 (2014).
[Crossref]

Chung, E.

Conkey, D. B.

Davidson, N.

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[Crossref]

DiMarzio, C. A.

Y. M. Wang, B. Judkewitz, C. A. DiMarzio, and C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat. Commun. 3, 928 (2012).
[Crossref]

Dudley, D.

D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14–25 (2003).
[Crossref]

Duncan, W. M.

D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14–25 (2003).
[Crossref]

Feld, M. S.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2, 110–115 (2008).
[Crossref]

Fink, M.

T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy, and S. Gigan, “Controlling light in scattering media noninvasively using the photoacoustic transmission matrix,” Nat. Photonics 8, 58–64 (2014).
[Crossref]

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012).
[Crossref]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[Crossref]

Friesem, A. A.

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[Crossref]

Gigan, S.

G. Volpe, G. Volpe, and S. Gigan, “Brownian motion in a speckle light field: tunable anomalous diffusion and selective optical manipulation,” Sci. Rep. 4, 3936 (2014).
[Crossref]

T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy, and S. Gigan, “Controlling light in scattering media noninvasively using the photoacoustic transmission matrix,” Nat. Photonics 8, 58–64 (2014).
[Crossref]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[Crossref]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley-Interscience, 2000).

Grabar, A. A.

Y. Liu, P. Lai, C. Ma, X. Xu, A. A. Grabar, and L. V. Wang, “Optical focusing deep inside dynamic scattering media with near-infrared time-reversed ultrasonically encoded (TRUE) light,” Nat. Commun. 6, 5904 (2015).

Grange, R.

Horstmeyer, R.

B. Judkewitz, Y. M. Wang, R. Horstmeyer, A. Mathy, and C. Yang, “Speckle-scale focusing in the diffusive regime with time reversal of variance-encoded light (TROVE),” Nat. Photonics 7, 300–305 (2013).
[Crossref]

Hsieh, C.-L.

Huignard, J.-P.

Huisman, T. J.

Jang, M.

Jayet, B.

Judkewitz, B.

M. Jang, H. Ruan, I. M. Vellekoop, B. Judkewitz, E. Chung, and C. Yang, “Relation between speckle decorrelation and optical phase conjugation (OPC)-based turbidity suppression through dynamic scattering media: a study on in vivo mouse skin,” Biomed. Opt. Express 6, 72–85 (2015).
[Crossref]

E. H. Zhou, H. Ruan, C. Yang, and B. Judkewitz, “Focusing on moving targets through scattering samples,” Optica 1, 227–232 (2014).
[Crossref]

H. Ruan, M. Jang, B. Judkewitz, and C. Yang, “Iterative time-reversed ultrasonically encoded light focusing in backscattering mode,” Sci. Rep. 4, 7156 (2014).
[Crossref]

B. Judkewitz, Y. M. Wang, R. Horstmeyer, A. Mathy, and C. Yang, “Speckle-scale focusing in the diffusive regime with time reversal of variance-encoded light (TROVE),” Nat. Photonics 7, 300–305 (2013).
[Crossref]

Y. M. Wang, B. Judkewitz, C. A. DiMarzio, and C. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat. Commun. 3, 928 (2012).
[Crossref]

Katz, O.

T. Chaigne, O. Katz, A. C. Boccara, M. Fink, E. Bossy, and S. Gigan, “Controlling light in scattering media noninvasively using the photoacoustic transmission matrix,” Nat. Photonics 8, 58–64 (2014).
[Crossref]

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[Crossref]

Lagendijk, A.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012).
[Crossref]

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[Crossref]

Lai, P.

Y. Liu, P. Lai, C. Ma, X. Xu, A. A. Grabar, and L. V. Wang, “Optical focusing deep inside dynamic scattering media with near-infrared time-reversed ultrasonically encoded (TRUE) light,” Nat. Commun. 6, 5904 (2015).

P. Lai, Y. Suzuki, X. Xu, and L. V. Wang, “Focused fluorescence excitation with time-reversed ultrasonically encoded light and imaging in thick scattering media,” Laser Phys. Lett. 10, 075604 (2013).
[Crossref]

P. Lai, X. Xu, H. Liu, and L. V. Wang, “Time-reversed ultrasonically encoded optical focusing in biological tissue,” J. Biomed. Opt. 17, 0305061 (2012).
[Crossref]

Lee, K.

Lerosey, G.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012).
[Crossref]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[Crossref]

Liebling, M.

Liu, H.

P. Lai, X. Xu, H. Liu, and L. V. Wang, “Time-reversed ultrasonically encoded optical focusing in biological tissue,” J. Biomed. Opt. 17, 0305061 (2012).
[Crossref]

X. Xu, H. Liu, and L. V. Wang, “Time-reversed ultrasonically encoded optical focusing into scattering media,” Nat. Photonics 5, 154–157 (2011).
[Crossref]

Liu, Y.

Y. Liu, P. Lai, C. Ma, X. Xu, A. A. Grabar, and L. V. Wang, “Optical focusing deep inside dynamic scattering media with near-infrared time-reversed ultrasonically encoded (TRUE) light,” Nat. Commun. 6, 5904 (2015).

C. Ma, X. Xu, Y. Liu, and L. V. Wang, “Time-reversed adapted-perturbation (TRAP) optical focusing onto dynamic objects inside scattering media,” Nat. Photonics 8, 931–936 (2014).
[Crossref]

Ma, C.

C. Ma, X. Xu, and L. V. Wang, “Analog time-reversed ultrasonically encoded light focusing inside scattering media with a 33,000 × optical power gain,” Sci. Rep. 5, 8896 (2015).
[Crossref]

Y. Liu, P. Lai, C. Ma, X. Xu, A. A. Grabar, and L. V. Wang, “Optical focusing deep inside dynamic scattering media with near-infrared time-reversed ultrasonically encoded (TRUE) light,” Nat. Commun. 6, 5904 (2015).

C. Ma, X. Xu, Y. Liu, and L. V. Wang, “Time-reversed adapted-perturbation (TRAP) optical focusing onto dynamic objects inside scattering media,” Nat. Photonics 8, 931–936 (2014).
[Crossref]

Mathy, A.

B. Judkewitz, Y. M. Wang, R. Horstmeyer, A. Mathy, and C. Yang, “Speckle-scale focusing in the diffusive regime with time reversal of variance-encoded light (TROVE),” Nat. Photonics 7, 300–305 (2013).
[Crossref]

Mosk, A. P.

Nixon, M.

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[Crossref]

Park, J.

Park, Y.

Piestun, R.

Popoff, S.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
[Crossref]

Psaltis, D.

C.-L. Hsieh, Y. Pu, R. Grange, and D. Psaltis, “Digital phase conjugation of second harmonic radiation emitted by nanoparticles in turbid media,” Opt. Express 18, 12283–12290 (2010).
[Crossref]

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2, 110–115 (2008).
[Crossref]

Pu, Y.

Ramaz, F.

Ruan, H.

Silberberg, Y.

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[Crossref]

Slaughter, J.

D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” Proc. SPIE 4985, 14–25 (2003).
[Crossref]

Small, E.

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[Crossref]

Suzuki, Y.

P. Lai, Y. Suzuki, X. Xu, and L. V. Wang, “Focused fluorescence excitation with time-reversed ultrasonically encoded light and imaging in thick scattering media,” Laser Phys. Lett. 10, 075604 (2013).
[Crossref]

Unser, M.

van Putten, E. G.

Vellekoop, I. M.

Volpe, G.

G. Volpe, G. Volpe, and S. Gigan, “Brownian motion in a speckle light field: tunable anomalous diffusion and selective optical manipulation,” Sci. Rep. 4, 3936 (2014).
[Crossref]

G. Volpe, G. Volpe, and S. Gigan, “Brownian motion in a speckle light field: tunable anomalous diffusion and selective optical manipulation,” Sci. Rep. 4, 3936 (2014).
[Crossref]

Vos, W. L.

Wang, L. V.

Y. Liu, P. Lai, C. Ma, X. Xu, A. A. Grabar, and L. V. Wang, “Optical focusing deep inside dynamic scattering media with near-infrared time-reversed ultrasonically encoded (TRUE) light,” Nat. Commun. 6, 5904 (2015).

C. Ma, X. Xu, and L. V. Wang, “Analog time-reversed ultrasonically encoded light focusing inside scattering media with a 33,000 × optical power gain,” Sci. Rep. 5, 8896 (2015).
[Crossref]

C. Ma, X. Xu, Y. Liu, and L. V. Wang, “Time-reversed adapted-perturbation (TRAP) optical focusing onto dynamic objects inside scattering media,” Nat. Photonics 8, 931–936 (2014).
[Crossref]

P. Lai, Y. Suzuki, X. Xu, and L. V. Wang, “Focused fluorescence excitation with time-reversed ultrasonically encoded light and imaging in thick scattering media,” Laser Phys. Lett. 10, 075604 (2013).
[Crossref]

P. Lai, X. Xu, H. Liu, and L. V. Wang, “Time-reversed ultrasonically encoded optical focusing in biological tissue,” J. Biomed. Opt. 17, 0305061 (2012).
[Crossref]

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J. Opt. Soc. Am. A (1)

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Supplementary Material (2)

NameDescription
» Supplement 1: PDF (2103 KB)      Supplemental Document
» Visualization 1: AVI (2671 KB)      Movie: continuous conjugate foci.

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Figures (7)

Fig. 1.
Fig. 1.

(a) Simplified schematic of the DMD-based DOPC. A 2.5 mm diameter collimated beam from the laser source (Excelsior 532 nm single mode, 200 mW, Spectra-Physics) is incident onto the sample through BS2. Scattered light from the sample is collected by L3 and is combined with the reference beam by BS3. The combined reference and sample beam is reflected by BS4 and Mirror 2, passes through BS4, and is captured by Camera 1 (pco.edge 5.5, PCO-TECH). The DMD (W4100, Wintech) and Mirror 2 are aligned symmetrically with reference to BS4, and the DMD surface is imaged onto the camera sensor chip by CL (AF-S VR Micro-NIKKOR 105 mm f/2.8G IF-ED, Nikon) with pixel-to-pixel alignment. Camera 1 and the DMD are connected through a host FPGA (ViClaro IV GX Camera Link Development Kit, Microtronix). The conjugate result is observed on Camera 2 (Prosilica GX 1920, Allied Vision) and the APD (SPCM-AQRH-14, Excelitas). (b) Optical path schematic of the recording step. (c) Optical path schematic of the playback step (L, lens; BS, beam splitter; BD, beam dump; CL, camera lens; APD, avalanche photodiode).

Fig. 2.
Fig. 2.

Functional schematic of the FPGA-based DOPC.

Fig. 3.
Fig. 3.

(a) DMD diffraction demonstration. (b) Binary phase modulation of a DMD. E p is the field played back by the DMD, E c is the desired phase conjugate field, and Δ θ is the phase difference between E c and E p . When a pixel of the DMD is turned on, it plays back the phasor E p , which can be decomposed into two orthogonal components. One is in the direction of the desired phase conjugate field E c with an amplitude modulated by cos | Δ θ | and contributes to the focus. The other component orthogonal to E c is modulated by sin | Δ θ | and contributes to the background.

Fig. 4.
Fig. 4.

Workflow of the FPGA-based DOPC.

Fig. 5.
Fig. 5.

(a) Moving sample setup. (b),(c) Moving sample speckle decorrelation curves at lateral velocity 0.2 and 0.25 mm/s. Error bars indicate standard deviation over 10 datasets. (d) Conjugate focus images and cross-section peak plots when the sample was static, moving at 0.2 and 0.25 mm/s.

Fig. 6.
Fig. 6.

PBR quantification. Scale bar is 100 μm.

Fig. 7.
Fig. 7.

(a) Clipped mouse dorsal skin setup and speckle decorrelation curve of in vivo tissue. Error bars indicate standard deviation over 10 datasets. (b) Continuous conjugate foci through clipped sample (see also Visualization 1). (c) APD plot for sustainable foci and decorrelation focus. Scale bar is 100 μm. In the experiment, to accurately distinguish the focus intensity from background in the APD signal, a background ( I b ) was measured first when the focus totally vanished, which is 5 × 10 5 in the APD plot.

Equations (6)

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I i ( x , y ) I ref ( x , y ) + 2 I ref ( x , y ) I sam ( x , y ) cos | Δ θ | .
I i ( x , y ) < I ref π 2 < | Δ θ | π , I i ( x , y ) > I ref 0 | Δ θ | π 2 .
d sin ϕ = n λ .
f ( | Δ θ | ) = { 0 , π 2 < | Δ θ | π cos | Δ θ | e i α + sin | Δ θ | e i β , 0 | Δ θ | π 2 .
PBR = 1 / 2 + ( N 1 ) / 2 π M N 2 π M ,
DMD ( x , y ) = { 1 , I i ( x , y ) < I ref ( x , y ) | α π | < ϕ ϕ = arccos ( 1 / 2 I ref / I sam ) < π 2 0 , else .

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